Problem: Solve for $x$ and $y$ using elimination. $\begin{align*}-4x-5y &= -1 \\ -7x-5y &= 5\end{align*}$
Answer: We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $-1$ and the bottom equation by $1$ $\begin{align*}4x+5y &= 1\\ -7x-5y &= 5\end{align*}$ Add the top and bottom equations. $-3x = 6$ Divide both sides by $-3$ and reduce as necessary. $x = -2$ Substitute $-2$ for $x$ in the top equation. $-4( -2)-5y = -1$ $8-5y = -1$ $-5y = -9$ $y = \dfrac{9}{5}$ The solution is $\enspace x = -2, \enspace y = \dfrac{9}{5}$.